Theorem prnz 4777

Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Hypothesis

Ref	Expression
prnz.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
prnz	⊢ {𝐴, 𝐵} ≠ ∅

Proof of Theorem prnz

Step	Hyp	Ref	Expression
1		prnz.1	. . 3 ⊢ 𝐴 ∈ V
2	1	prid1 4762	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3	2	ne0ii 4344	1 ⊢ {𝐴, 𝐵} ≠ ∅

Syntax hints:   ∈ wcel 2108   ≠ wne 2940  Vcvv 3480  ∅c0 4333  {cpr 4628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-un 3956  df-nul 4334  df-sn 4627  df-pr 4629

This theorem is referenced by:  opnz  5478  propssopi  5513  fiint  9366  fiintOLD  9367  wilthlem2  27112  upgrbi  29110  wlkvtxiedg  29643  shincli  31381  chincli  31479  constrextdg2lem  33789  spr0nelg  47463  sprvalpwn0  47470